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Full text of "Handbook Of Chemical Engineering - I"

MECHANICAL SEPARATION
283
In a cylindrical tank the horizontal velocity along or parallel to the axis of X can
RV be obtained from the proportion V: vx = x : R or vx ~----the velocity of a particle
at any distance x from the center of the tank.
Now -^ = vx =  and xdx
RvdL
												1			I	>	^f^
	398												^p *	s*	-JB		<<
	51											X	^		X"	"	
	141									^	/?		s	^ \	[>		
	63.1								dJ>	/		[^					
>								y	c		X*						
	17.8							:/		/ff							
u. o   .	10.						J		:/								
I							^	.	/;								
^-ss <								/;									
						j		/:									
	1.					'/	/'										
	TWtf				f	.y	I										
	.316				/	^											
	.178			/		{											
0	.1		/		/												
9 MANTISSAS OF LOG. 0
FIG. 13.Fall of quartz and galena.
x2 Integrating both sides of the equation V^ becomes equal to RVt or t = ^y which
gives the time for any particle to advance towards the overflow horizontally a distance x. In time t, however, the particle has subsided the distance mt or y. Substituting  for t in the above time expression and transposing, the equation for path
is derived as already given.
The various formulas given enable a comparison between circular and rectangular tanks on the point of capacity. Having determined the rate of subsidence of the particle which will reach the bottom at the periphery of a circular tank of any diameter by dividing the depth of water by time, t = ^^Tp the factor is obtained which will give the time for the same particle to traverse a rectangular tank of any cross section which has the same rate of feeding as the circular tank. It is only necessary to divide the factor into the proper depth of water of the rectangular tank to obtain this figure. The velocity of flow in the rectangular tank is obtained by dividing the rate of feeding by the cross sectional area of the water. This figure times the traversing time will give the length of rectangular tank equal in settling capacity to the circular one.
On these computations being performed for any hypothetical pair of tanks the areas of the two tanks are found to differ so slightly that for all practical purposes it can be stated that settling tanks of equal area have the same settling capacity. It is impossible to compute the exact head of discharge in each case because of the effect of the velocity of approach, the theory of correction for which is only an approximation and this question has not been considered in the discussion. In the case of the